Abstract:If {\bf T} is a complete theory stronger than {\bf ZF}$_{\hbox {Fin}}$ such that axiom of extensionality for classes + {\bf T} + $(\exists X)\Phi _i$ is consistent for 1$\leq i \leq k$ (each alone), where $\Phi _i$ are normal formulae then we show {\bf AST} + $(\exists X)\Phi _1 +...+ (\exists X)\Phi _k$ + scheme of choice is consistent. As a consequence we get: there is no proper $\Delta _1$-formula in {\bf AST} + scheme of choice. Moreover the complexity of the axioms of {\bf AST} is studied, e.g. we show axiom of extensionality is $\Pi _1$-formula, but not $\Sigma _1$-formula and furthermore prolongation axiom, axioms of choice and cardinalities are $\Pi _2$-formulae, but not $\Pi _1$-formulae in {\bf AST} without the axiom in question.
Keywords: alternative set theory, complexity of formulae, $\Pi _2$-formula, extension of axiomatic systems
AMS Subject Classification: Primary 03E70; Secondary 03H15, 03A05, 03D55